Satisfying Solution

In my previous post, I posed the problem:

Problem Let and be polynomials in two variables such that and exist. Show that the determinant

is not a nonzero constant.

Dennis gave a nice, if brief solution, and Pace fleshed it out a bit in an e-mail to me. I now understand where the mysterious “trace operator” in the solution comes from, and that’s what I want to explain here.

First, Dennis’ solution. For any polynomial , define . We now observe the mysterious identity

.

Now, if contains no positive powers of , then $\sum f_{(i+m)i$ is zero for positive . Similarly, the hypothesis on forces the third factor to be zero for negative . And, when , the in front kills the summand. So the conditions on and force to be zero. In particular, is not a nonzero constant.

It’s very slick. The only question is, what is ? The explanation, as it turns out, gives us a nice chance to practice our differential forms.

Without futher ado, the answer.

.

That is, we are averaging the function over the unit disc. I always get confused when ‘s show up, so let me just emphasize that, although we are talking about a complex valued function on the complex plane (namely ), this is a perfectly ordinary integral over a disc.

We can check this quickly in polar coordinates:

.

So, why is ? Consider as a map from one -dimensional vector space to another. I want to distinguish the domain and the range, so I’ll write . I’ll use the coordinates on and on . Consider the two form on .

A reassuring sidenote: is a smooth (complex valued) function on , so there is no reason we can’t take its differential as a function on . Later, we’ll see ‘s showing up, and there is nothing deep about them either. The definitions of differential forms and integrals which you learned in real analysis are still valid in complex analysis. (If you wonder why I make such a point of this, remember that differential operators do mean something different; try explaining the computation of in terms of small movements along a path!)

In any case, notice that . Indeed, this is almost how the Jacobian is defined. So is the integral of over the disc in . (Up to the factor of , and the ratio between and .)

But is closed, so is closed as well. So I can move that integral to any disc with the same boundary. In particular, if is well defined, then I can push the integral onto the disc . But is a holomorphic disc of one complex dimension! So the pull back of to , along a holomorphic map, is a wedge of two proportional -forms. In short, it must vanish.

When and do not obey the above limiting conditions, then we can’t push the integrating surface off to infinity, as there is a pole near the origin of . I leave it to the reader to see why we get the no-longer-mysterious identity above in that setting. Hint: isn’t just closed, it’s also exact.

I’ll close by wondering whether there is any good theory of discs like ; discs in such that the restriction of is the area form. For example, are such discs determined by their boundary? Which curves bound such discs?

About these ads